Theorem crnggrpd 20194

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20193	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20189	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18875  CRingccrg 20181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-ring 20182  df-cring 20183

This theorem is referenced by:  fermltlchr  21496  psdmul  22121  psd1  22122  psdpw  22125  ply1chr  22262  ply1fermltlchr  22268  elrgspnsubrunlem1  33341  elrgspnsubrunlem2  33342  erlbr2d  33358  rlocaddval  33362  rloccring  33364  rloc0g  33365  rlocf1  33367  fracerl  33400  gsumind  33438  ressply1evls1  33658  evl1deg1  33669  evl1deg2  33670  evl1deg3  33671  ply1dg1rt  33673  vr1nz  33686  psrmonprod  33729  mplmonprod  33731  esplyfvn  33754  irngss  33865  extdgfialglem1  33870  irredminply  33894  algextdeglem4  33898  algextdeglem5  33899  aks6d1c1p3  42480  aks6d1c2lem4  42497  aks6d1c6lem2  42541  aks5lem2  42557  evlsbagval  42927  selvvvval  42943  evlselv  42945  selvadd  42946  prjcrv0  42991